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We study expected runtimes for quantum programs. Inspired by recent work on probabilistic programs, we first define expected runtime as a generalisation of quantum weakest precondition. Then, we show that the expected runtime of a quantum program can be represented as the expectation of an observable (in physics). A method for computing the expected runtimes of quantum programs in finite-dimensional state spaces is developed. Several examples are provided as applications of this method, including computing the expected runtime of quantum Bernoulli Factory -- a quantum algorithm for generating random numbers. In particular, using our new method, an open problem of computing the expected runtime of quantum random walks introduced by Ambainis et al. (STOC 2001) is solved.

Quantum Variational Circuits (QVCs) are often claimed as one of the most potent uses of both near term and long term quantum hardware. The standard approaches to optimizing these circuits rely on a classical system to compute the new parameters at every optimization step. However, this process can be extremely challenging both in terms of navigating the exponentially scaling complex Hilbert space, barren plateaus, and the noise present in all foreseeable quantum hardware. Although a variety of optimization algorithms are employed in practice, there is often a lack of theoretical or empirical motivations for this choice. To this end we empirically evaluate the potential of many common gradient and gradient free optimizers on a variety of optimization tasks. These tasks include both classical and quantum data based optimization routines. Our evaluations were conducted in both noise free and noisy simulations. The large number of problems and optimizers yields strong empirical guidance for choosing optimizers for QVCs that is currently lacking.

We present a $9^k\cdot n^{O(1)}$-time algorithm for the proper circular-arc vertex deletion problem, resolving an open problem of van 't Hof and Villanger [Algorithmica 2013] and Crespelle et al. [arXiv:2001.06867]. Our structural study also implies parameterized algorithms for modification problems toward proper Helly circular-arc graphs.

Copy-protection allows a software distributor to encode a program in such a way that it can be evaluated on any input, yet it cannot be "pirated" - a notion that is impossible to achieve in a classical setting. Aaronson (CCC 2009) initiated the formal study of quantum copy-protection schemes, and speculated that quantum cryptography could offer a solution to the problem thanks to the quantum no-cloning theorem. In this work, we introduce a quantum copy-protection scheme for a large class of evasive functions known as "compute-and-compare programs" - a more expressive generalization of point functions. A compute-and-compare program $\mathsf{CC}[f,y]$ is specified by a function $f$ and a string $y$ within its range: on input $x$, $\mathsf{CC}[f,y]$ outputs $1$, if $f(x) = y$, and $0$ otherwise. We prove that our scheme achieves non-trivial security against fully malicious adversaries in the quantum random oracle model (QROM), which makes it the first copy-protection scheme to enjoy any level of provable security in a standard cryptographic model. As a complementary result, we show that the same scheme fulfils a weaker notion of software protection, called "secure software leasing", introduced very recently by Ananth and La Placa (eprint 2020), with a standard security bound in the QROM, i.e. guaranteeing negligible adversarial advantage. Finally, as a third contribution, we elucidate the relationship between unclonable encryption and copy-protection for multi-bit output point functions.

The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation stating that arbitrarily long quantum computations can be performed with a polylogarithmic overhead provided the noise level is below a constant level. A recent work by Fawzi, Grospellier and Leverrier (FOCS 2018) building on a result by Gottesman (QIC 2013) has shown that the space overhead can be asymptotically reduced to a constant independent of the circuit provided we only consider circuits with a length bounded by a polynomial in the width. In this work, using a minimal model for quantum fault tolerance, we establish a general lower bound on the space overhead required to achieve fault tolerance. For any non-unitary qubit channel $\mathcal{N}$ and any quantum fault tolerance schemes against $\mathrm{i.i.d.}$ noise modeled by $\mathcal{N}$, we prove a lower bound of $\max\left\{\mathrm{Q}(\mathcal{N})^{-1}n,\alpha_\mathcal{N} \log T\right\}$ on the number of physical qubits, for circuits of length $T$ and width $n$. Here, $\mathrm{Q}(\mathcal{N})$ denotes the quantum capacity of $\mathcal{N}$ and $\alpha_\mathcal{N}>0$ is a constant only depending on the channel $\mathcal{N}$. In our model, we allow for qubits to be replaced by fresh ones during the execution of the circuit and we allow classical computation to be free and perfect. This improves upon results that assumed classical computations to be also affected by noise, and that sometimes did not allow for fresh qubits to be added. Along the way, we prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude damping noise resolving a conjecture by Ben-Or, Gottesman, and Hassidim (2013).

We give a quantum algorithm for solving the Bounded Distance Decoding (BDD) problem with a subexponential approximation factor on a class of integer lattices. The quantum algorithm uses a well-known but challenging-to-use quantum state on lattices as a type of approximate quantum eigenvector to randomly self-reduce the BDD instance to a random BDD instance which is solvable classically. The running time of the quantum algorithm is polynomial for one range of approximation factors and subexponential time for a second range of approximation factors. The subclass of lattices we study has a natural description in terms of the lattice's periodicity and finite abelian group rank. This view makes for a clean quantum algorithm in terms of finite abelian groups, uses very relatively little from lattice theory, and suggests exploring approximation algorithms for lattice problems in parameters other than dimension alone. A talk on this paper sparked many lively discussions and resulted in a new classical algorithm matching part of our result. We leave it as a challenge to give a classcial algorithm matching the general case.

We consider symmetric positive definite preconditioners for multiple saddle-point systems of block tridiagonal form, which can be applied within the MINRES algorithm. We describe such a preconditioner for which the preconditioned matrix has only two distinct eigenvalues, 1 and -1, when the preconditioner is applied exactly. We discuss the relative merits of such an approach compared to a more widely studied block diagonal preconditioner, specify the computational work associated with applying the new preconditioner inexactly, and survey a number of theoretical results for the block diagonal case. Numerical results validate our theoretical findings.

Fitting geometric models onto outlier contaminated data is provably intractable. Many computer vision systems rely on random sampling heuristics to solve robust fitting, which do not provide optimality guarantees and error bounds. It is therefore critical to develop novel approaches that can bridge the gap between exact solutions that are costly, and fast heuristics that offer no quality assurances. In this paper, we propose a hybrid quantum-classical algorithm for robust fitting. Our core contribution is a novel robust fitting formulation that solves a sequence of integer programs and terminates with a global solution or an error bound. The combinatorial subproblems are amenable to a quantum annealer, which helps to tighten the bound efficiently. While our usage of quantum computing does not surmount the fundamental intractability of robust fitting, by providing error bounds our algorithm is a practical improvement over randomised heuristics. Moreover, our work represents a concrete application of quantum computing in computer vision. We present results obtained using an actual quantum computer (D-Wave Advantage) and via simulation. Source code: //github.com/dadung/HQC-robust-fitting

We contribute to fulfil the long-lasting gap in the understanding of the spatial search with multiple marked vertices. The theoretical framework is that of discrete-time quantum walks (QW), i.e. local unitary matrices that drive the evolution of a single particle on the lattice. QW based search algorithms are well understood when they have to tackle the fundamental problem of finding only one marked element in a $d-$dimensional grid and it has been proven they provide a quadratic advantage over classical searching protocols. However, once we consider to search more than one element, the behaviour of the algorithm may be affected by the spatial configuration of the marked elements, due to the quantum interference among themselves and even the quantum advantage is no longer granted. Here our main contribution is twofold : (i) we provide strong numerical evidence that spatial configurations are almost all optimal; and (ii) we analytically prove that the quantum advantage with respect to the classical counterpart is not always granted and it does depend on the proportion of searched elements over the total number of grid points $\tau$. We finally providing a clear phase diagram for the QW search advantage with respect to the classical random algorithm.

Learning a graph topology to reveal the underlying relationship between data entities plays an important role in various machine learning and data analysis tasks. Under the assumption that structured data vary smoothly over a graph, the problem can be formulated as a regularised convex optimisation over a positive semidefinite cone and solved by iterative algorithms. Classic methods require an explicit convex function to reflect generic topological priors, e.g. the $\ell_1$ penalty for enforcing sparsity, which limits the flexibility and expressiveness in learning rich topological structures. We propose to learn a mapping from node data to the graph structure based on the idea of learning to optimise (L2O). Specifically, our model first unrolls an iterative primal-dual splitting algorithm into a neural network. The key structural proximal projection is replaced with a variational autoencoder that refines the estimated graph with enhanced topological properties. The model is trained in an end-to-end fashion with pairs of node data and graph samples. Experiments on both synthetic and real-world data demonstrate that our model is more efficient than classic iterative algorithms in learning a graph with specific topological properties.